Since the 19th century, it has been gradually uncovered a deep and rich relation between asymptotic question in number theory, for instance, the number of primes less than a given magnitude, and Fourier analysis. This relation is embodied in the Riemann Zeta and L-functions that form a fascinating class of analytic functions in which many mysteries of number theory are buried.

The two talks aim at the construction of the Riemann Zeta functions and Dirichlet L-functions based on the theory of Mellin transforms. We use as main reference Tate's thesis that has been published in Cassels and Frohlich's volume "Algebraic number theory"

14h00-16h00: D-modules and Bernstein-Sato polynomials (Phung Ho Hai)

Abstract:

Mellin transform was originally defined for functions on the positive half line. Given a smooth function on the set of positive real numbers that decays rapidly as the parameter tends to zero or infinity, its Mellin transform is well-defined as a holomorphic function on the whole complex plane. One is interested in the Mellin transform of some classes of functions that do not necessarily decay. The Mellin transform of those functions are meromorphic functions, which are primarily only defined on a certain right half plane of the complex plane. The analytic continuation of these function to the whole complex plane is achieved by a trick of partial integration.

I.M. Gelfand made a conjecture that the above construction can be generalized to the case, where the positive half line is replaced by a connected component of the non-zero locus in the n-dimensional real space of a polynomial in n variables (in the classical case, the polynomial is P(x)=x). This conjecture was first proved by M. Atiyah, then Gelfand's student J. Bernstein gave a pure algebraic proof using D-modules. Briefly speaking, the partial integration technique mentioned above was taken care by the celebrated Gelfand-Sato polynomial (differential operator) and its existence follows from the holonomy of a certain D-module.

The aim of my two talks is first to explain Bernstein's proof as a motivation to D-module theory and then to develop some basic facts about D-modules. The main reference is Bravermann lecture notes.